“Car Body Concept Modeling: Congruency Equations for Hybrid Structures”

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OTTORATO DI

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ICERCA IN

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ROGETTO E

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VILUPPO DI

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RODOTTI E

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ROCESSI

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NDUSTRIALI

Car Body Concept Modeling:

Congruency Equations for Hybrid Structures

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Background and aim of the work 3

D

OTTORATO DI

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ICERCA IN

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ROGETTO E

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VILUPPO DI

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RODOTTI E

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ROCESSI

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NDUSTRIALI

CICLO XXV

Car Body Concept Modeling:

Congruency Equations for Hybrid Structures

Settore Scientifico Disciplinare ING/IND 14

Candidato

Ing. Alessio Moroncini ___________________

Tutore Controrelatore

Prof. Niccolò Baldanzini Prof. Domenico Mundo

____________________ ____________________

Coordinatore del Dottorato

Prof. Marco Pierini ____________________

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Abstract

The aim of this PhD thesis is to define a set of congruency equations for the rigorous connection of dissimilar FEM elements for hybrid thin-walled structures, whereby one-dimensional elements are coupled with three-dimensional joint structures.

The set of congruency equations defined within the framework of this research activity has been applied in the field of automotive concept modeling.

The first part of the thesis concerns with an introduction to the car body concept modelling method for the early design phase of the “Body in White” with a special focus on NVH structural optimization and “hybrid modeling” techniques.

The second part of the thesis relates to the definition of the connecting equations for the hybrid structures through the “principle of virtual work” and a fundamental property of the center of mass of cross sections defined by the author within this dissertation.

Finally a validation phase of the congruency equations for hybrid structures and the integration of a hybrid model in a multi-objective optimization sequence, is presented.

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1 INTRODUCTION

The continuous research activity on high performance simulation models and methods, seeking for an always higher accuracy and reliability, has driven this study towards the definition of a set of congruency equations for the rigorous connection of dissimilar FEM elements for concept modeling applications. In this first chapter, the background, the aim of this work and an introduction to the “vehicle concept modeling” methodology for car body design, context of this work, is described.

1.1 Background and aim of the work

The present research activity has been fully endorsed by the European Commission through a Marie Curie Action. The project, “VEhicle Concept Modeling” (VECOM), has been joined by 14 academic and industrial partners across Europe with the aim of providing dedicated research activities and trainings in the emerging field of vehicle concept modelling for up-front pre-CAD functional performance engineering, bridging between industry and academia. In particular this dissertation has been developed at BMW Group in Munich, Germany (acting as host organization), under the academic supervision of the University of Florence, Italy. The “vehicle concept modeling” research area is of highly strategic importance to European automotive OEMs, who must launch products on an ever shorter time frame, at increased quality of multiple performance attributes. When simulation results become available in an early design stage, problems can already be solved before the first detailed CAD model is created, which will increase the quality of the first detailed simulation models and reduce the time to market. Moreover, early what-if studies can be performed to balance and optimize possibly conflicting performance attributes (safety, NVH, dynamics, durability ...) at an increased feasibility and at reduced costs [1].

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In an effort to shorten development times, concept modeling CAx-methods play an important role in automotive industry. In so-called upfront CAE processes, concept methods are introduced earlier in the functional car development phase. The goal is to realize as early as possible a full vehicle concept, satisfying all functional performance requirements, such as for vibration and acoustics comfort. Very high costs for late concept changes and costly investigations on hardware prototypes are therefore avoided.

The modeling process comprises the assessment of different concept variations from the very early design stages up to detailed construction and optimization in the pre-series and series phases.

Using concept modeling applications, the designer can evaluate a large number of solutions and define, in a relative short time, the most suitable structure capable of meeting the structural targets within the defined design space.

In the early design phase concept methods are very powerful for suitability and feasibility studies, however they are characterized by a lack of accuracy and robustness, a main drawback that relegates them to a more qualitative rather than quantitative design/research techniques.

The aim of this research activity is therefore to provide a more powerful car body concept modeling method, seeking for higher accuracy and robustness, reducing development time and costs.

The most critical area of the concept modeling approach is clearly the characterization of the joints, a new joint modeling solution has been proposed introducing the so called “FEM hybrid structures”, whereby one-dimensional FE elements are coupled with three-dimensional joint structures and morphing techniques are applied to the three-dimensional parts in order to perform a full-body geometrical optimization process (see paragraph 4.1).

The implementation of hybrid structures in the concept model is a complex task, in particular the definition of congruency equations for the rigorous connections of the dissimilar elements (1D-3D) is a major engineering challenge and represents nevertheless the ultimate target of this Ph.D dissertation.

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Introduction to the vehicle concept modeling 11

1.2 Introduction to the vehicle concept modeling

Concept modeling FEM design processes are well known modeling techniques and widely applied in the early design phase of any complex structure that must fulfill a set of global multi-disciplinary and high-performance structural requirements.

The key idea behind the concept modeling approach is to create a simplified car body model with a certain degree of parameterization, keeping in mind that it should be “as coarse as possible, as detailed as necessary”. The simplification of the model is normally realized by reducing the number of nodes and introducing geometrically simplified elements (mono-dimensional FE elements), while the parameterization can be realized in different ways depending on the software in use (i.e. introducing specific parameterized elements, using external solver, etc…). The reduced number of nodes assures a shorter calculation time while the parameterization defines the optimization variables.

Besides the concept modeling commercial software, like “SFE concept” or dedicated modules integrated in the CAE-driven development software (e.g. CATIA), the mainstream concept modelling approach is based on the combination of mono- dimensional thin walled parameterized beams and two-dimensional shell elements, defining the three dimensional structures. The car body structure is virtually divided in beam-like parts, carrying the loads, and panels. The structural parts are modeled using mono-dimensional beam-like elements with assigned bi-dimensional cross section properties, while the panels are modeled using shell-like elements, whereby a coarsening process is used to reduce the mesh discretization to the level of the mono-dimensional beam-like elements.

The car body concept modeling, based on combination of mono and bi-dimensional elements, is a well-established design process, so called "beams and shells” concept model and it is used for early concept phase car body structural optimization with a special focus on the NVH performance [2] [3] [4] [5].

The idea behind the “beam and shell” concept modeling is to transform as much as possible geometrical information of the FE model into properties information in order to create a model fast to calculate and easy to parameterize, that means easy to optimize.

An FE-model for the early concept phase has different requirements compared to one for the series development. Typically detailed CAD-geometry informations are not necessary and even not available. In this context of lacking geometric information and of demanding for quick answers in a short time, the simulation with so-called “beam and shell” FE models has proven to be a very effective process. The purpose of the investigations based on the “beam and shell” model is to evaluate the potential of different structural geometrical and topological

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solutions in order to achieve premium NVH performance with minimum weight and construction space, while matching a complete set of vehicle customer relevant functional requirements, also including basic requirements for ride, handling and crash.

A fine FE model of a vehicle body usually consists mainly of 2D shell elements. The use of shell elements is appropriate where one dimension is small compared the other two dimensions, these elements are characterized by the fact that only two dimensions are modeled by nodes, while the third dimension (i.e. the thickness) is specified as a property. Considering that a body structure consists mainly of joined metal parts, a fine FE shells model is a perfect solution for a detailed representation, however not really suitable for topological optimizations in the early design phase.

In the early vehicle concept phase, where clear structural performance targets are set (dynamic stiffness, static stiffness, crash performance, acoustic and vibration comfort performance, etc.), but geometrical information are limited (wheel base, car width), the “beam and shell” model proves to be effective for global car body structural investigations and optimizations. An example of a BMW “Body in White” (BiW) concept model with arbitrary beam cross sections is shown in figure 1.

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Concept modeling “state of the art” 13

1.3 Concept modeling “state of the art”

Design processes, especially in the concept design phase, are part of the core know-how of any engineering company, hence defining the “prior art”, defining the “state of the art” while positioning the BMW “beam and shell” concept model, is not an easy task. A meaningful way to place the BMW modeling approach with respect of the prior art is to evaluate individually the single technical features of the model itself.

The key technical features characterizing a concept model can be summarized as follows:

 Type of mono-dimensional thin walled beam model

 Type and number of variables

 Type of constraints

 Type of load cases

 Type of optimization strategy

 Type of connections between beam and bi-panel elements

 Type of joint modeling Considering that:

 The “beam and shell” model is based on arbitrary beam cross section mono-dimensional beams with 7DOF (for more information, see paragraph 1.5), and that these are the most advanced mono-dimensional FEM beams available in commercial software (in literature, models with higher number of DOF are available, however they are defined just for a restricted type of cross sections and still under validation);

 The “beam and shell” model has more than 1500 variables describing completely the geometry of the BiW, (large scale optimization problem), normally in the automotive applications a maximum of 300 variables are used;

 Constraints and load cases of the “beam and shell” model are the same applied in the development of the static and dynamic BiW structural performance during the final phase of design at BMW;

 The optimization strategy of the “beam and shell” model (see description in chapter 2) has been recently developed and tested within the VECOM project, providing best results in concept modeling applications;

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 The connections between beam and bi-dimensional panel elements are rigid standard connections and specifically designed to emulate with a minimum error the structural behavior of the welding spots.

It is possible to state that the “beam and shell” model represents the “state of the art” in the field of concept modeling for static and dynamic investigations. The solution to the problem posed, lack of accuracy and robustness, has to be found in the definition of the joint modeling approach, clearly playing a dominant role in this context.

A general description and a critical review of the car body design and optimization process is presented in the upcoming paragraphs. The focus is on the latest joint modeling solutions, introduced to increase the robustness and the accuracy of the modeling process (see chapter 3).

1.4 Concept model construction

The main idea behind the creation of a “beam and shell” model (BS model), is to describe the geometry as much as possible through properties of elements instead of the actual geometrical coordinates of the nodes of the elements. Using full shell-FE models only plate thicknesses are available for optimization, whereby the use of mono-dimensional beam elements allows a parametric description of the complete beam cross section, using beam width, height and the respective wall thicknesses.

To start the creation of the BS model, see figure 2, the designer selects a suitable starting model ranging from just sketches, a full CAD model, a predecessor fine FE model, a predecessor BS model or a mixture thereof. Basically it depends on whether a brand new car concept is being developed or whether it is more an evolutionary development of a predecessor.

The geometry of the selected starting model is then first subdivided into regions for beam modeling, consisting of the main load carrying structure of the body, and the shell-like rest of the structure, consisting of the body panels. All the beams are modeled using mono-dimensional elements with assigned bi-dimensional cross section properties. The rest of the body is kept as shells elements, where a coarsening process is used to reduce the mesh discretization to the level of the mono-dimensional beam elements. The beam and shell elements are then coupled together using an in-house rigid-connections coupling scheme forming the final FE concept model.

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Concept model construction 15

When modeling from scratch, i.e. based on a limited set of dimensions and sketches, the mono-dimensional beam elements have to be generated in dedicated standard CAE pre-processors. More often however the modeling starts from existing shell element models, whereby the mono-dimensional beam elements and their section properties can be largely automatically generated by cutting the shell elements along the beam-like structure (an in-house tool has been developed for this process but equivalent commercial tools are also available). Based on the cut section geometry, the tool generates the actual nodes and elements across the centers of gravity of the section cuts of the beam-like structure, evaluates the section properties and the actual geometry of the beam cross-section in term of line segments. If necessary this line segment geometry information can then be imported in dedicated cross section editors for further simplification and modification. Once all the beam cross sections have been defined, the generated beam properties can be assigned by the user to the beam elements in the FE preprocessor.

Using the FEM standard property definition, the actual geometry of the cross section is no longer available and therefore not accessible during the optimization process. In an iterative optimization sequence these geometrical properties are therefore converted using an in-house tool to equivalent properties, in terms of cross sectional area and moments of inertia that can be edited.

The FEM solver beam library includes a series of basic geometric shapes such as e.g. rectangular, U-profile, L-profile, I-profile, etc. In general closed sections will be modeled with rectangular-type sections and open sections with U-profile type sections. When using open sections the important cross section ‘warping’ effects will be taken into account. The property entries for the FEM solver standard sections consist of dimensions describing the standard basic cross section geometry, such as width, height and wall thicknesses. These dimensions can be directly addressed as design variables in the optimization sequence.

In recent years however, a more sophisticated approach to model beam cross sections has been introduced, the so called “Arbitrary Beam Cross Sections” (ABCS) method, allowing an exact geometric definition of the beam cross section. Two different geometry descriptions are available. For the so-called GS-sections, the cross section geometry is described by the area defined of an outer-loop and a series of inner-loops.

The second definition is the so-called center-line description, suitable for thin-walled structures, whereby a series of line segments describes the medial line of the cross-section and wall thicknesses can be assigned on the level of each line segment. The bounding box height and width of these ABCS cross sections, as well as the wall thickness for the center line sections, are available as design variables for the optimization sequence. A more detailed description of the method is provided in section 3.

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Figure 2 Workflow for the creation of beams and shells FE concept models

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A very important modeling aspect is the connection of the beams in the different car body structure joints, where three or more beams are to be connected. Empirical rules have been defined at BMW to model these joints with a combination of open- and closed beam sections based on the judgment of the actual 3D-geometry

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Concept model construction 17

of the joints and the presence of internal lateral or longitudinal stiffening plates inside the joint structure.

Figure 3 Joint between upper B-pillar, upper side frame and middle roof cross beam

In figure 3, the modeling of the joint between upper B-pillar, the upper side frame beam and the middle cross roof beam is shown. Especially the joints connecting the A-, B-, C-pillars to the rest of the structure play an important role in global car body stiffness and should be modeled with care.

The remaining panel structure of the car body structure is meshed with shell elements with a mesh discretization equal to the size of the beam elements. The beam and shell structure is then merged together and connected through rigid connectors using an in-house connection scheme. To finalize the BS concept model a series of special points are set using an in-house numbering scheme identifying the relevant load, constraint and response positions, enabling standardized automatic definition of the different load cases. The beam and shell elements of the car body are then subdivided in predefined logical groups according to the different car body parts. These logical groups can be addressed for global constraints settings for the optimization sequence (see next section), as well as for added mass distribution for setting up trimmed body FE concept models. A set of standardized trace lines is defined along the beam-like structure of the body, enabling automatic post-processing and visualization of the results. The model is then finally subjected to standard calculation runs, including modal analysis and global static bending and torsion load case, for a model quality check [4] [5].

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1.5 ABCS beams

In this section the advantages of the implementation of the arbitrary beam cross sections applied on the BMW “beam and shell” models, are discussed.

In general, in commercial FEM software three different types of mono-dimensional beam models are available for concept modeling applications:

 1D beam model with equivalent cross section: no geometrical information included.

 1D beam model with equivalent standard cross section selected from a library (C, T, circular shape etc..).

 1D beam with arbitrary beam cross section, ABCS.

The big drawback of using the equivalent cross sections or the limited set of the equivalent standard cross sections from the beam library is the total loss of the actual cross section geometrical information. After the optimization only statements on the level of cross sectional geometric properties can be done. The link to the actual optimized cross section geometry is no longer available, as shown in figure 4, [2] [4].

The use of ABCS beams allows for geometrically exact description of any real beam cross sections, described through property cards.

The ABC-sections can be defined through a “general” and through a “center line” section description. For both descriptions the bounding box width and height can be defined as design variables in order to realize a 2D scaling of the real cross section. The center line description is particularly suitable for thin-walled cross sections. Using this description it is not only possible to allow for 2D bounding box scaling, but also the thicknesses of wall segments are available as design variables. The arbitrary cross section shown is an ABCS beam section of type center line. An additional advantage of the ABCS description is the possibility to explicitly handle longitudinal internal stiffeners in the optimization sequence. In the standard section approach these stiffeners ‘disappear’ in the ‘equivalent’ section properties and are consequently not directly accessible through a design variable in the optimization sequence.

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ABCS beams 19

Figure 4 Cross section of a rocker panel modeled using a standard section (left) and an arbitrary cross section (right).

The benefits of running an optimization process with the ABCS beams compared to the standard sections of the beam library can therefore be summarized as follows:

 Direct optimization of the real cross section geometry: not only qualitative but also quantitative information is available to improve the body structure in terms of bounding box size and wall thicknesses.

 Description and optimization of sections with longitudinal stiffeners is only possible with ABCS beams .

 Higher acceptance of the results by the designers. The designer recognizes his beam construction.

 No conversion to equivalent standard cross section necessary The introduction of the ABCS beams provides considerably more flexibility for design changes in the different optimization cycles. The overall 2D scaling of the beam cross section bounding box in the optimization sequence can indeed still be seen as a limitation. However it is considered as an important step towards full parametric beam cross section modeling and optimization.

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1.6 Multi-attribute optimization of the concept model

Once the complete concept model has been created, a set of 1500 FEM variables and a multi-objective optimization function are generated, the target of the optimization process is to minimize the weight, investigate potential construction space, while matching the global relevant customer requirements. A representation of the full optimization process, including the model build up phase, is displayed in figure 5, [5].

Figure 5 Representation of the optimization process including the model build-up phase

The optimization functions include 25 different load cases, each load case has a different weighting factor that can be weighted depending on the design requirements. The 25 load cases can be grouped in 5 main categories:

 Static stiffness

 Dynamic stiffness

 Roll over (linearized)

 Steering column vibration

 Pseudo front crash (linearized)

The aim of the investigations is to ensure premium NVH performances for the “Body in White” in different operational conditions. The most important study case is the dynamic stiffness one, however a multi objective approach has been developed in order to ensure, already in the concept phase, the achievement of all

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Multi-model optimization 21

the global structural performance, including crash, avoiding late and expensive design modifications.

1.6 Multi-model optimization

An interesting aspect of the beam and shell concept model is that can be easily applied nowadays for parallel multi-model optimizations. The always higher degree of standardisation of the body in white components required by the OEMs to keep low the manufacturing costs while assuring high-quality standards, drives the designers to implement a modular design approach. This approach leads to an always higher number of components shared between different car bodies of the same line-up (and sometimes even of crossed line-up). In figure 6 an example of the possible common components shared among three different car bodies of the same line-up is shown. Once the concept models of the three car bodies are created, the same set of optimization parameters can be assigned to the shared components (in orange) of the three different car bodies. Different optimization targets for each vehicle can be defined, according to the different structural requirements, and finally a single optimization function for the unique set of design variables of the shared components can be defined. Finally a parallel FEM calculation of the bodies can be executed. The great advantage of a multi-model concept optimization is that a set of common shared parts can be optimized at the same time for the different structural requirements of several vehicles.

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Optimization strategies: general overview 23

2 OPTMIZATION STRATEGY

In this section, a brief overview of the strengths and weaknesses of genetic and gradient-based algorithms, in the context of optimization problems whereby each function evaluation is relatively expensive, is given. The focus is on problems in which the number of design variables is signiﬁcantly greater than the number of objectives and constraints, more details on the optimization strategy and penalty method implemented in the “beam and shell” model are also given.

2.1 Optimization strategies: general overview

In figure 7 a scheme gives a general overview of the possible optimzation strategies applicable for BiW optimization sequences.

In gradient based strategies, the cost of the optimization can be considered roughly proportional to the number of design variables. Rapid convergence is the primary advantage of a gradient-based method. Clearly, proper exploitation of gradient information can signiﬁcantly enhance the speed of convergence in comparison with a method that does not compute gradients. Another feature of gradient-based methods is that they provide a clear convergence criterion. If the gradient is reduced by many orders of magnitude, one can be conﬁdent that at least a local optimum has been reached.

One of the key disadvantages of gradient-based methods is the “development” cost. Whether the linearization is performed by hand or using automatic differentiation, with a complex code this can be time-consuming. Another potential weakness of gradient-based methods is that they are relatively intolerant of difﬁculties such as noisy objective function spaces, inaccurate gradients, categorical variables, and topology optimization. Another oft-mentioned disadvantage of gradient-based methods is that they ﬁnd a local rather than a global optimum.

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However, in many engineering design contexts this is unlikely to be an issue, since the highly constrained nature of the design problem inhibits multimodality. The key disadvantages of gradient-based methods are precisely the strengths of genetic algorithms. First, genetic algorithms treat the function evaluation as a “black box”. Consequently, development cost is minimal. Second, they are tolerant of noise in the objective function and have no difﬁculty with categorical variables or topology changes. Furthermore, in principle, genetic algorithms ﬁnd a global optimum. The key disadvantage associated with genetic algorithms is that they can converge very slowly, especially near an optimum. A second weakness is that determining a termination criterion is not straightforward.

It is not yet well understood how well either a gradient-based or a genetic algorithm can deal with such design problems. Although a genetic algorithm can proceed in principle, insufficiently converged solutions can produce misleading results that can lead the genetic algorithm into nonoptimal areas of the design space. The above strengths and weaknesses must be considered in choosing an optimization algorithm for a specific problem class. A key tradeoff is between the relatively high development cost of the gradient-based method using an adjoint formulation and the relatively high computational cost of the genetic algorithm. The more frequently the algorithm is to be used, the more beneficial the gradient-based algorithm becomes. Clearly, a quantitative assessment of the computational cost of the two algorithms is needed to make an intelligent choice for a given class of problems [6][7][8][9][10].

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2.2 Optimization strategy of the concept model 25

2.2 Optimization strategy of the concept model

The “beam and shell” multi-disciplinary vehicle optimization process is built around a gradient based solver.

A gradient based solver has been preferred over a genetic algorithm because of the rapid convergence, the drawback of finding just local minima is partially counter-measured launching the gradient based solver optimization starting from different initial design configurations, up to ten times, increasing consistently the chances of approaching a global minima.

The setup of the optimization model includes the identification of the design variables, setting-up the design constraints and the optimization cost function. For the optimization sequence of the car body a series of relevant load cases are taken into account for optimal full vehicle NVH as well as ride and handling performance. These cases include a series of global static bending and torsion load cases, car body eigen-modes, frequency response functions at customer relevant locations and local static stiffness at chassis connection points. These levels of static stiffness will indeed directly influence the accuracy of the steering response and the overall driving experience, as the body is acting as a spring between the two axes during driving maneuvers. On top of that a pseudo crash load case is included, in order to take into account minimal crash requirements for the car body in an approximate way as well. For all these load cases specific targets are set for the car body, based on detailed analysis of customer relevant functional performance requirements for the full vehicle. The objective of the optimization is to find an ‘optimal’ car body structure with minimal weight, satisfying the different load case targets and constraints within the available construction space of the car body. For the complete study of a body structure up to 1500 design variables are taken into account while considering an equivalent number of constraint equations.

A tailored penalty method so called ß-Method (see paragraph 2.4) is also included in the objective function in order to increase the chance of getting optimization results into the feasible design area [2] [4].

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2.3 Creating the optimization model

The setup of an optimization model of about 1500 variables is a challenging and important modeling phase, it includes the identification of the design variables, setting-up the design constraints and the optimization cost function and requires a high degree of automation. In this paragraph the automated approach developed for the “beam and shell” model is described [2].

Figure 8 OptiCenter GUI for setting-up design variables and constraints

The objective of the optimization is to find an “optimal” car body structure with minimal weight, satisfying the different load case targets and constraints within the available construction space of the car body.

The available construction space of the car body is defined by setting the beam cross section geometrical parameters – height, width and wall thickness – as design variables with suitable min/max limits, based on full vehicle package and design, technological and manufacturing (e.g. minimal sheet metal thickness, width/height ratio of beam structures, etc.) constraints. For the complete study of a

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2.3 Creating the optimization model 27

body structure up to 1500 design variables are taken into account while considering an equivalent number of constraints equations.

To facilitate the definition of this large set of design variables and constraints an in-house tool so called OptiCenter, see figure 8, has been developed. Based on the imported FEM bulk data file of the concept model and the predefined logical groups the design variables are automatically identified and design limits are set based on predefined default settings or specific user input. The corresponding FEM cards are created and exported into the input deck.

Using the same tool the different load cases to be considered can be selected and necessary target values can be specified by the user as shown in figure 9. For the global dynamic car body stiffness up to four eigen-modes can be targeted during the optimization. The user provides the target frequency level, the mode shape type (i.e. 1. bending, 1. torsion, 2. bending, etc.) and the mode number necessary for the mode tracking during the optimization iterations. Furthermore a minimal frequency difference between two modes can be targeted, e.g. 3 Hz between first global modes.

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The identification of the targeted modes are evaluated in a nominal run, the so-called check-run, where all load cases are evaluated for the starting configuration of the concept model.

Regarding the static stiffness, a series of standardized equivalent static bending, torsion and lateral stiffness load cases have been defined for the car body and handled in the multi-load case optimization sequences. Target values, as well as the location where the static stiffness values have to be evaluated, identified in the nominal check run, are specified by the user. The target static stiffness values are converted to static deformation levels in the specified locations.

Next to these main load cases, targeting overall optimal car body stiffness, a series of pseudo frontal and rollover crash load cases are taken into account, ensuring minimal crash requirements for the car body. The frontal crash load case is set-up as an inertial relief calculation, whereby the car body is deformed by the inertial forces induced by the crash impact forces. These pseudo crash and rollover load cases are targeted by specifying maximum allowable strain values in the beam structure of the green house of the car body.

More specific vibration comfort load cases can be specified as well in terms of frequency response monitoring. Steering wheel vibration levels are for example monitored in the relevant frequency range of the stationary regime of the engine.

2.4 The penalty method: ß-method

In the gradient based optimization sequence constraints are satisfied first, before minimizing the objective function. In the beta-method, the monitoring of the different load case targets is formulated as a constraint and introduced into the objective function through a newly introduced design variable (β), measuring the constraint violation.

These design variables so-called beta-values are combined with the car body weight into the objective function, whereby the relative importance of all load cases can be balanced in the optimization cost function by specifying relative weighting factors.

The above mentioned constraints are formulated in such a way that minimizing the beta-value implies reaching the specified target values. In the equation (1) the constraint equation is shown for targeting the eigen-frequency of a tracked mode as a minimal value [2].

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2.4 The penalty method: ß-method 29

1 )

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et t st et t

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(1)

The same procedure can be done also in case that a variable should be below a certain maximum value or should match an exact target.

When the beta-method is used, a limit curve (ftarget) is defined first, which is used for the normalization of the frequency response (λ1st). With the help of the

additionally introduced design variable β, whose starting value is usually one, the constraints are defined, the design objective becomes a function of the design variable β and a minimization or maximization of objective function can only be achieved through a minimization of maximization of β value. An important point is that introducing the beta-method several subcases can be optimized at the same time.

Without the beta-method, once the constraints have been violated the gradient-based solver would stop before minimizing the objective function, in order to prevent this situation, the constraints are “relaxed” through the additional variable beta and the objective function is minimized/maximized in function of the beta values, avoiding the termination the optimization sequence, reaching the target values.

The beta-method optimization sequence is based on a gradient based optimizer, this means that the optimized solution is found step-by-step in the direction of the steepest descent, identified by evaluating the sensitivities of all responses with respect to all design variables. A common problem with this type of optimizers is the convergence to local minima. No guarantee can be given on whether a global optimum has been found. A common procedure is therefore to re-start the optimization sequence from different initial configurations, in an effort to explore the full design space and therefore increase the probability to find a more global optimum. These different initial configurations can either be specified by the user or can also be generated randomly by perturbation of the design variables from the nominal design within the given limits.

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2.5 Post-processing of the results

The gradient based optimization sequence for these coarse “beam and shell” FE concept models, handling well over 1500 design variables, generates a large amount of result data in the form of binary (op2) and ASCII files. Dedicated in-house post-processing tools have been developed in order to interpret these data in a comprehensive and standardized way.

Key-results are automatically extracted and presented in tables, graphs and contour plots in a dedicated intranet, allowing for quick reporting and results comparison when studying different car body concepts. Figure 10 shows the optimization history of the first bending (green line) and torsion (red line) mode versus their target value. Similar plots are generated for all static load cases, as well as, for the evolution of the car body weight to be minimized.

Three-dimensional contour plots of the full model are also generated with extensive information on the change in construction space, i.e. change of beam cross section width and height, and the change of beam cross section wall thicknesses. Figure 12 shows an example of a contour plot, depicting the change of the outer dimensions of the different beam elements. Such contour plots give a good overview on the areas of car body structure where constructive measures preferably have to be taken in order to achieve the set targets in an optimal way [2].

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2.5 Post-processing of the results 31

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2.6 Application case

In order to provide the reader with a better overview on concept modeling applications, a practical example of how the beams and shells concept modeling process has been recently used at BMW during the early concept development phase of new car projects is presented. The following type of design questions were to be analyzed:

 What are the structural modifications for a station wagon car body needed to fulfill significantly increased functional static and dynamic stiffness targets, while obeying different limitations concerning construction space?

 Can the proposed structural modifications be realized in the current design or is a new structural topology needed?

Figure 11Change in wall thicknesses after an optimization of a station wagon car body structure with limited construction space

In figure 11 the optimized carrier structure of the reference middle class station wagon is shown with the optimization restriction that the existing construction space should not be exceeded. The colors show the scaling of the wall thickness with respect to the initial values. In this optimization run, the targeted dynamic stiffness could not be reached, even though wall thicknesses have been increased to up to 300 percent of the original values in some regions. Further increasing wall thicknesses clearly results more in an increase of mass than an

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2.6 Application case 33

increase in car body stiffness, leading to a loss of dynamic stiffness and lower global eigen-frequency levels. Through the use of the earlier described beta-method, the optimization continued converging to the best-compromise solution while residing in an infeasible design region, not completely fulfilling the eigen-frequency target constraints.

Figure 12Change in construction space after an optimization of a station wagon car body structure without limitations for the construction space.

The scale reported in figure 12 is related to the relative change in construction space, i.e. the outer dimensions of the carrier structure after an optimization process where an almost unlimited construction space scaling is allowed. Special restrictions are set for the longitudinal front and rear carrier that are not allowed to shrink because of crash requirements and for the upper A-pillar, which is not allowed to increase due to design reasons.

With those boundary conditions, a significant increase of about 200 up to 700 percent in some areas of the car body structure can be observed. Especially for the roof carriers and the B-pillar the needed construction space increases enormously. The resulting extremely large carrier cross sections are not acceptable for package, design and ergonomic reasons. Further optimization runs with more realistic restrictions on the outer carrier dimensions have lead again to best-compromise results without reaching the dynamic eigen-frequency target values.

From these examinations the conclusion was drawn, that the car body structure of this middle class station wagon is not very well suited as base for the new considerably increased dynamic stiffness targets. The increased functional requirements clearly call for an alternative carrier topology [2].

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Strengths and weaknesses analysis 35

3 CRITICAL REVIEW OF THE CONCEPT

MODEL

In this chapter a critical review of the “beam and shell” model based on “strength and weaknesses” analysis supported by simulation data is presented. The focus is on the most critical part of the BiW concept model: the joints. The actual joint modeling approach, the “state of the art” of the joint modeling and a new solution are also presented.

3.1 Strengths and weaknesses analysis

The “beam and shell” model for the early design phase of the body in white presents, according to the opinion of the author and of a group of experienced engineers in the field of concept modeling (a survey has been conducted among seven experienced engineers), the following strengths and weaknesses:

Strengths:

 Short calculation and optimization time.

 Multi-model optimization: parallel optimization of common parts, shared in different vehicles with different structural targets.

 High degree of geometrical details in the concept phase: the full geometry of the cross-sections is maintained (ABCS).

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Weaknesses:

 Low modeling accuracy due to inconsistent joint modeling approach.

 Long modelling time.

 Density correction factor needed in order to take into account the loss of mass due to the lack of minor components (e.g. door hinges, plastic sub-frames etc..).

 Not yet suitable for composite materials.

3.2 Modeling error

The beam and shell model is a useful design tool in the early car design phase for feasibility studies, the model has been developed in house for years, parallel to a wide set of dedicated pre and post processing tools. A detailed list of rules and design guide lines are also provided and made available on-line internally. Despite those efforts, the modeling error is relatively high and the modeling process has a low level of repeatability.

It is well known among modeling experts that the most critical part of the “beam and shell” model design process is the characterization of the joints, the global static and dynamic stiffness of the car body depends mostly on the right modeling of them. A set of tests have been done in order to identify all the possible source of error of the model, taking into account several modeling aspects, the results confirmed the dominant role played by the modeling of the joints. Minor changes on the modeling of the joints (within the BMW standards and guide lines) could increase noticeably the modeling error, up to the double of the expected values.

In the graph of figure 13, the different deformations lines of the full body of a BMW E90 are described. The red lines describe the deformation lines of the reference model (fine FE model) under certain loads (load related information are confidential) versus six different concept models realized by six different designers following the internal standard and modeling guidelines. The six concept models reported in figure 13 differs from each other mainly for the modeling of the joints, however the resulting models show quite a significant difference between each other (a quantitative comparison is not provided due to confidentiality). In other words each designer, deriving a concept joint form the reference fine model, has to “interpret” each joint thus resulting in a non-consistent quality of the concept model, strongly dependent on the experience of the designer.

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Modeling error 37

Additionally the quality of the current beam and shell concept model has been assessed comparing a fine FE model of a BMW E90 (reference model) with a “beam and shell” model directly derived from the same fine model. The two models have been investigated under same load cases and the result cross-checked with the measured data of an E90 BiW available from the BMW measurement center, see figure 14.

More in detail, the comparison was based on two different static load cases, bending and torsion, the results from the modal analysis have been also compared, first bending and first torsion eigen-modes.

Figure 13 Deformation lines of different E90 BiW models

Figure 14 Models and data analyzed for the evaluation of the quality of the “beam and shell” model

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From these two load cases, 10 different values describing the static stiffness of the BiW are calculated and compared (e.g. global bending stiffness, front local stiffness green house). For a full understanding and description of the deformation status of the model also five trace lines have been defined and compared. The total mass, the position of the center of gravity and principal axes of inertia have been also compared.

The result of assessment can be summarized as follow:

 The total mass, principal axis of inertia and the center of gravity of the concept model differs from the reference model of ~1%.

 The global static stiffness in bending case is forecasted within an error of ~5%.

 The global static stiffness in torsion case is forecasted within an error of ~8%.

 The local stiffness is forecasted with an error depending of the different cases, max: 30%.

 The error of the dynamic stiffness is evaluated through a MAC matrix and is about 5%.

Additional modeling aspects that are worth to mention are related to the welding spots and flanges. The effect of the distribution of the welding spots is unfortunately not taken into account in the concept model, however considering that the concept model is in general quite diverse from the final production model and that it is used mainly for qualitative studies, this is not a critical aspect.

Regarding the flanges, the cross section of the mono-dimensional beam of the concept model are modeled without flanges, reducing in this way the moments of inertia of the sections, this it is done in order to compensate the augmented stiffness due to the fact that the mono-dimensional beam are completely closed along the longitudinal axes, because, as already mentioned, the welding spots are not modeled.

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3.3 Joint modeling: actual situation 39

3.3 Joint modeling: actual situation

In order to provide a better understanding of the actual joint modeling approach used for the realization of the “beam and shell” model, in this paragraph a short description of the method, using practical examples, is provided.

Each joint of the concept model is modeled using a combination of mono-dimensional beams with open or closed profiles depending on the actual geometry and topology of the reference joint. For each type of joint, the designer has to decide, following the guidelines and the internal standard, which combination of profiles need to be used in order to characterize in the most accurate way the joint, taking also into account eventual reinforcement plates.

For instance in figure 15 and in figure 16 the guidelines for modeling two different joints connecting the B-pillar with the rocket panel are shown. The figure 15 indicates that in case the joint has three reinforcement plates, it needs to be modeled using beams element with closed cross sections in the three directions, while in figure 16, in case the joint has two reinforcement plates, it should be modeled using a combination of one beam with closed profiles and the other two with open profiles. All the geometrical parameters of the beams in red are parameterized and therefore can be optimized.

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Figure 15 Example of modeling guideline for a joint modeling in the “beam and shell model” with three reinforcement plates

Figure 16 Example of modeling guideline for a joint modeling in the “beam and shell model” with two reinforcement plates

The main asset of the current modeling approach is the possibility of optimize the full body without losing completely the related geometrical information, however the price to pay is a lower accuracy, due to the “interpretation” of each joint that the designer as to perform.

In the following paragraphs a “state of the art” of the joint modeling and a new modeling approach combining a high calculation accuracy while keeping the geometrical optimization applied to joints and other complex structures is presented.

T-joint: rigid configuration

T-joint: soft configuration

Reinforcement plates Reinforcement plates

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3.4 State of the art of the joint modeling 41

3.4 State of the art of the joint modeling

In order to find a solution for a more robust joint modelling approach, the “state of the art” of the joint modeling has been defined.

In literature fifteen different solutions have been found, selected and evaluated according to the following criteria [11][12][13][14]:

 Modeling time

 Calculation time (SOL200)

 Stress calculation (possible yes/not)

 Geometrical information

 Repeatability

 Accuracy

 Need of external solver (for optimization)

 Innovation degree

 Optimization information (geometrical information available after optimization)

All the concept joint modeling approaches found in literature are more or less sophisticated implementation of:

 super-elements

 spring-joint representations

The super-elements and spring-joints models have two main advantages: accuracy and repeatability.

The super-elements have been also tested on the “beam and shell” model confirming the expected accuracy.

However they have two main drawbacks: total loss of geometrical information during the modeling phase and no chance to perform stress calculation.

Due to the above mentioned drawbacks, both solutions, the super-elements and the spring-joint modelling, have been judged “not suitable” to be implemented in the beam and shell model.

A new joint modelling approach based on hybrid structures, whereby mono-dimensional beam-like elements are coupled with three-dimensional detailed joint structures, providing accuracy and repeatability while keeping the needed geometrical information and the possibility of calculating the stress of the structures

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has been implemented, a detailed description of this solution is given in the upcoming paragraph.

A summary of the evaluation of the modeling solutions including the new proposed solution is shown in figure 17.

All the joint modeling approaches have been assessed referring to the actual BMW method based on a combination of beams with open and closed arbitrary cross sections. At the BMW method has been assigned the value 0 for each criteria assuming it as the reference, each criteria of each model has been evaluated following the scale below.

SCALE

2 This modeling method has a clear advantage regarding this aspect 1 This modeling method has a light advantage regarding this aspect

0 No clear advantages or disadvantages

-1 This modeling method has a light disadvantage regarding this aspect -2 This modeling method has a clear disadvantage regarding this aspect

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3.5 The new joint modeling approach 43

3.5 The new joint modeling approach

The new joint modeling solution proposed in the context of this dissertation is based, as already anticipated, on the hybridization of the concept model, whereby mono-dimensional beam-like FE elements are coupled with three-dimensional detailed joint structures and morphing techniques (see paragraph 3.6) are applied to the three-dimensional parts in order to realize a complete body geometrical optimization, see figure 18, the red arrows indicate the direction of motion of the control points of the morphing box.

The three-dimensional joint structures can be designed starting from the available information: sketches, full CAD models, a predecessor fine FE model, or a mixture thereof. Using FE shell elements just the thickness of the elements are available as optimization parameters, in order to overcome this problem, a morphing box is applied to the joint shell model and morphing parameters can be used as optimization control parameters. In this way a precise calculation and an optimization of the geometry can be performed in parallel.

The hybridization of the model has the potential to achieve the requested targets in terms of robustness and accuracy, while the usage of detailed shell-like structure combined with morphing applications leave the opportunity of performing geometrical multi-disciplinary optimization [4].

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The new proposed concept model is named “Hybrid Morphing Concept Model”, it represents a more “detailed” concept modeling approach, aspects like stress calculation, geometrical optimization, connecting elements between dissimilar elements, have been investigated in order to make possible his integration in the design and optimization process.

The hybridization process has been applied to fourteen selected joints, see figure 19, playing key roles on the static and dynamic stiffness calculation.

Figure 19 Selected joints for the hybridization process

Clearly the overall hybridization process increases the number of nodes of the concept model, but a relative small increase of details has the potential to assure a remarkable increase of accuracy. In figure 20 a graph shows the benefit of the hybridization process in terms of global stiffness calculation.

The current concept model with about 10000 nodes, in worst case, leads to an error on the calculation of the global stiffness up to 12% (in average 8%). Applying the hybridization, arriving to ~20000 nodes, the global error can be kept under 5% (error related to the fine FE model). Looking at pure numbers, the hybridized model has the double of nodes of the actual concept model, however from a pure computational point of view, it is not a critical aspect, the model remains anyway quick to run and to optimize. The new trend proposed, in the concept modeling design, is to increase the degree of detail, where necessary, according with the always higher calculation power available.

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3.5 The new joint modeling approach 45

Figure 20 Global stiffness calculation error vs number of nodes of the concept model

The advantages of introducing a new hybrid concept model in the design process can be summarized as follow:

- Increase the robustness of the modeling process.

- Increase the structural performance calculation accuracy in the early design phase.

- Provide more geometrical information on the optimized joints in order to achieve the design targets.

Worst case 10000; 12% 20000; 5% 1500000; 0% 0 2 4 6 8 10 12 14 10000 100000 1000000 e rr or (% )

number of nodes (log10)

Global stiffness calculation error

FE Fine model Concept model

! ! Based on one single e90 concept model ! !

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3.6 Morphing

In this paragraph a summary of the morphing techniques available in commercial software is presented.

Morphing techniques can be grouped in two main categories: the so called “box morphing” and the “direct morphing”, the main difference consists in that in the first case, morphing actions are controlled using control points, in the second case, using design variables [15].

 The Box Morphing method has three different approaches. Box Morphing - Approach A

Multiple morphing boxes that follow the shape of the structure. Moving or sliding of control points results in the morphing of the model in the desired direction. This approach allows the user to slide one part on another, or reshape a part by moving the control points of the boxes.

Box Morphing - Approach B

A single morphing box, split into many, with their edges fit on the feature lines of the model.

This approach has the following advantages:

- the surrounding boxes act as buffer zones of the morphing action, thus ensuring continuity of the deformed neighboring morphed entities.

- the ability to move the fit morphing box edges by exact translations, rotations or even snapping onto predefined target 3D curves, allows for highly controllable and precise modifications of the loaded entities. This approach is recommended for CFD models, but also for structural assemblies, when a modification of a part affects the surrounding components also.

Box Morphing - Approach C

A morphing box can handle the shape of other boxes. This approach has the following advantages:

- separate groups of morphing boxes can handle different features of the same model, without the need of complex morphing boxes.

- local and global modifications can be done of a model easily without the need of complicated script commands. This approach is recommended when local (detailed) and global modifications are needed in the same model.

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3.6 Morphing 47

 Direct Morphing

FE-models morphing can also be performed without morphing boxes. This can be performed either by specifying frozen/rigid nodes and morphing zones, or origin and target curves. Creating local depressions on shell mesh is also possible. This method is suggested for local modifications of a part.

The selection of the most suitable morphing technique takes into consideration mainly two factors:

- modeling time

- coupling with the optimizer

Considering that the modeling time is similar for each approach, and that the joints can be optimized independently without interact with the rest of the structures (e.g. no need for the more complex approaches B or C), the possible choices are either approach A either direct morphing.

Considering also that the “beam and shell” model is optimized using Nastran SOL200, and that ANSA, the pre-processor in use, supports the Manual Grid

Variation Method of NASTRAN SOL200, the best morphing approach results on being

the direct morphing. The approach selected supports simultaneous optimization in different directions: multiple vectoring per point and every vector can be controlled with an own design variable.

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Introduction to hybridization 49

4 MODEL HYBRIDIZATION

In the previous chapter a new joint modeling approach, based on hybrid structures has been presented. The focus of this chapter is to provide the reader with a general overview on hybrid structures and more inside information on the engineering challenge of the homogenous stress calculation in structures made by dissimilar FEM elements.

4.1 Introduction to hybridization

Nowadays the most accurate method to investigate the structural performance of the thin-walled structures, like BiW, is based on detailed tridimensional shell-like structures, up to 3.000.000 nodes. In the concept phase however parameterized one-dimensional beams elements, replacing the shell-like elements, are widely used, the overall computational analysis results much quicker: ~10.000 nodes, see figure 20. Parameterized concept models give to the designer great freedom in the optimization phase, having full access to the geometrical parameters of the structure, investigating, in very short time, several topological solutions of the load-carrying structural members.

The accuracy of the concept model is far from the shell-like detailed model, the main error derives from the joint modeling based on the one-dimensional beam, and secondary from the one-dimensional model (7 DOF) of the beams [4].

Within this research activity a new concept model is propose and it is based on hybrid structures, whereby not only one-dimensional thin-walled beam elements, but also shell-like thin-walled detailed structures, are allowed. The one-dimensional FE beam elements are now coupled with three-one-dimensional detailed joint structures (shell-like elements) and morphing techniques are applied to the three-dimensional parts in order to realize a full body geometrical optimization, see

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figure 19. The target is to increase accuracy and robustness of the actual concept model. The one-dimensional beam model used in the “beam and shell” concept model is a 7 DOF arbitrary beam cross-sections model proposed by Nastran. The beam model itself shows a good accuracy in static conditions and some limitations in dynamic analysis (local dynamic).

The introduction of shell-like elements in the concept model brings back the problem of the parameterization of the geometry of the model in order to launch optimization runs, while mono-dimensional elements provide a complete set of parameters to be optimized (height, length and thickness), in the shell-like elements just the thickness can be parameterized and therefore optimized, in order to perform a complete geometrical optimization of the shell-like three-dimensional structure of the joint, morphing techniques have been selected and applied for the complete geometrical optimization of shell-like structures, where else just the thicknesses of the shell elements would have been accessible for the optimization sequence.

Morphing techniques find a wide use in several fields, for finite element applications two main different morphing approaches are available: the so called “box morphing” and the “direct morphing”.

The “box morphing” approach is based on multiple morphing boxes that follow the shape of the structure. Moving or sliding of control points, results in the morphing of the model in the desired direction. See figure 21, in red the control points.

The “direct morphing” approach defines an origin, a target shape and the relative deformation vectors to move from the origin to the target. The final shape is controlled by design variables. See figure 22 in green, the deformation vectors.

“Car Body Concept Modeling: Congruency Equations for Hybrid Structures”

D

OTTORATO DI

R

ICERCA IN

P

ROGETTO E

S

VILUPPO DI

P

RODOTTI E

P

ROCESSI

I

NDUSTRIALI

Car Body Concept Modeling:

Congruency Equations for Hybrid Structures

D

OTTORATO DI

R

ICERCA IN

P

ROGETTO E

S

VILUPPO DI

P

RODOTTI E

P

ROCESSI

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NDUSTRIALI

Car Body Concept Modeling:

Congruency Equations for Hybrid Structures

Abstract

Contents

1

INTRODUCTION

1.1

Background and aim of the work

1.2

Introduction to the vehicle concept modeling

1.3

Concept modeling “state of the art”

1.4

Concept model construction

1.5

ABCS beams

1.6 Multi-attribute optimization of the concept model

1.6

Multi-model optimization

2

OPTMIZATION STRATEGY

2.1

Optimization strategies: general overview

2.2 Optimization strategy of the concept model

2.3 Creating the optimization model

2.4 The penalty method: ß-method

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2.5 Post-processing of the results

2.6 Application case

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